Let $F$ be a free group (I'm interested in the case $F$-finitely generated).
Let $\gamma_n(F)$ be the corresponding terms of the lower descending series, so for example $\gamma_4(F)=[F,[F,[F,F]]]$.
Let $F^"$ be the following term of the derived series $F^"=[[F,F],[F,F]]$.
The 3-subgroups lemma gives $F^"\subseteq\gamma_4(F)$.
How much is known about the failure of the previous statement for $\gamma_5$ instead of $\gamma_4$? Can someone give me some references about $$\frac{\gamma_5}{\gamma_5\cap F^"}$$ or $$\frac{F^"}{\gamma_5\cap F^"}$$
Are they infinite? Are they finitely generated (for F-finitely generated)?
[Edit:] It's pretty hard to Google this question, but I'm sure someone, somewhere, did it, unless it's obvious but I can't see the answer.
$F''/(\gamma_5 \cap F'') \cong F''\gamma_5/\gamma_5$ is finitely generated because it is a subgroup of the finitely generated nilpotent group $F/\gamma_5$. Since it is nontrivial, and $F/\gamma_5$ is torsion-free, it must be infinite.
But $\gamma_5/(\gamma_5 \cap F'') \cong F''\gamma_5/F''$ is not finitely generated, because $F'/(F''\gamma_5)$ is finitely generated (again because it's a subgroup of a finitely generated nilpotent group) but $F'/F''$ is not.